!$Id$ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !all function below are taken from math.f90 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! module math real*8, parameter :: pi = 3.14159265358979323846264338327950288419716939937510 ! *** 3x3 Identity *** real*8, dimension(3,3), parameter :: math_I3 = & reshape( (/ & 1.0,0.0,0.0, & 0.0,1.0,0.0, & 0.0,0.0,1.0 /),(/3,3/)) contains !************************************************************************** ! matrix multiplication 33x33 = 3x3 !************************************************************************** pure function math_mul33x33(A,B) implicit none integer i,j real*8, dimension(3,3), intent(in) :: A,B real*8, dimension(3,3) :: math_mul33x33 forall (i=1:3,j=1:3) math_mul33x33(i,j) = & A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) return end function math_mul33x33 !************************************************************************** ! Cramer inversion of 3x3 matrix (subroutine) !************************************************************************** PURE SUBROUTINE math_invert3x3(A, InvA, DetA, error) ! Bestimmung der Determinanten und Inversen einer 3x3-Matrix ! A = Matrix A ! InvA = Inverse of A ! DetA = Determinant of A ! error = logical implicit none logical, intent(out) :: error real*8,dimension(3,3),intent(in) :: A real*8,dimension(3,3),intent(out) :: InvA real*8, intent(out) :: DetA DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )& - A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )& + A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) if (DetA <= tiny(DetA)) then error = .true. else InvA(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA InvA(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA InvA(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA InvA(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA InvA(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA InvA(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA InvA(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA InvA(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA InvA(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA error = .false. endif return END SUBROUTINE math_invert3x3 !******************************************************************** ! determinant of a 3x3 matrix !******************************************************************** pure function math_det3x3(m) implicit none real*8, dimension(3,3), intent(in) :: m real*8 math_det3x3 math_det3x3 = m(1,1)*(m(2,2)*m(3,3)-m(2,3)*m(3,2)) & -m(1,2)*(m(2,1)*m(3,3)-m(2,3)*m(3,1)) & +m(1,3)*(m(2,1)*m(3,2)-m(2,2)*m(3,1)) return end function math_det3x3 !**************************************************************** pure subroutine math_pDecomposition(FE,U,R,error) !-----FE = R.U !**************************************************************** implicit none real*8, intent(in) :: FE(3,3) real*8, intent(out) :: R(3,3), U(3,3) logical, intent(out) :: error real*8 CE(3,3),EW1,EW2,EW3,EB1(3,3),EB2(3,3),EB3(3,3),UI(3,3),det error = .false. ce = math_mul33x33(transpose(FE),FE) CALL math_spectral1(CE,EW1,EW2,EW3,EB1,EB2,EB3) U=DSQRT(EW1)*EB1+DSQRT(EW2)*EB2+DSQRT(EW3)*EB3 call math_invert3x3(U,UI,det,error) if (.not. error) R = math_mul33x33(FE,UI) return end subroutine math_pDecomposition !************************************************************************** ! Cramer inversion of 3x3 matrix (function) !************************************************************************** pure function math_inv3x3(A) ! direct Cramer inversion of matrix A. ! returns all zeroes if not possible, i.e. if det close to zero implicit none real*8,dimension(3,3),intent(in) :: A real*8 DetA real*8,dimension(3,3) :: math_inv3x3 math_inv3x3 = 0.0 DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )& - A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )& + A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) if (DetA > tiny(DetA)) then math_inv3x3(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA math_inv3x3(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA math_inv3x3(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA math_inv3x3(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA math_inv3x3(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA math_inv3x3(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA math_inv3x3(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA math_inv3x3(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA math_inv3x3(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA endif return end function math_inv3x3 !********************************************************************** ! HAUPTINVARIANTEN HI1M, HI2M, HI3M DER 3X3 MATRIX M !********************************************************************** PURE SUBROUTINE math_hi(M,HI1M,HI2M,HI3M) implicit none real*8, intent(in) :: M(3,3) real*8, intent(out) :: HI1M, HI2M, HI3M HI1M=M(1,1)+M(2,2)+M(3,3) HI2M=HI1M**2/2.0-(M(1,1)**2+M(2,2)**2+M(3,3)**2)/2.0-M(1,2)*M(2,1)-M(1,3)*M(3,1)-M(2,3)*M(3,2) HI3M=math_det3x3(M) ! QUESTION: is 3rd equiv det(M) ?? if yes, use function math_det !agreed on YES return END SUBROUTINE math_hi !********************************************************************** pure subroutine math_spectral1(M,EW1,EW2,EW3,EB1,EB2,EB3) !**** EIGENWERTE UND EIGENWERTBASIS DER SYMMETRISCHEN 3X3 MATRIX M implicit none real*8, intent(in) :: M(3,3) real*8, intent(out) :: EB1(3,3),EB2(3,3),EB3(3,3),EW1,EW2,EW3 real*8 HI1M,HI2M,HI3M,TOL,R,S,T,P,Q,RHO,PHI,Y1,Y2,Y3,D1,D2,D3 real*8 C1,C2,C3,M1(3,3),M2(3,3),M3(3,3),arg TOL=1.e-14 CALL math_hi(M,HI1M,HI2M,HI3M) R=-HI1M S= HI2M T=-HI3M P=S-R**2.0/3.0 Q=2.0/27.0*R**3.0-R*S/3.0+T EB1=0.0 EB2=0.0 EB3=0.0 IF((ABS(P).LT.TOL).AND.(ABS(Q).LT.TOL))THEN ! DREI GLEICHE EIGENWERTE EW1=HI1M/3.0 EW2=EW1 EW3=EW1 ! this is not really correct, but this way U is calculated ! correctly in PDECOMPOSITION (correct is EB?=I) EB1(1,1)=1.0 EB2(2,2)=1.0 EB3(3,3)=1.0 ELSE RHO=DSQRT(-3.0*P**3.0)/9.0 arg=-Q/RHO/2.0 if(arg.GT.1) arg=1 if(arg.LT.-1) arg=-1 PHI=DACOS(arg) Y1=2*RHO**(1.0/3.0)*DCOS(PHI/3.0) Y2=2*RHO**(1.0/3.0)*DCOS(PHI/3.0+2.0/3.0*PI) Y3=2*RHO**(1.0/3.0)*DCOS(PHI/3.0+4.0/3.0*PI) EW1=Y1-R/3.0 EW2=Y2-R/3.0 EW3=Y3-R/3.0 C1=ABS(EW1-EW2) C2=ABS(EW2-EW3) C3=ABS(EW3-EW1) IF(C1.LT.TOL) THEN ! EW1 is equal to EW2 D3=1.0/(EW3-EW1)/(EW3-EW2) M1=M-EW1*math_I3 M2=M-EW2*math_I3 EB3=math_mul33x33(M1,M2)*D3 EB1=math_I3-EB3 ! both EB2 and EW2 are set to zero so that they do not ! contribute to U in PDECOMPOSITION EW2=0.0 ELSE IF(C2.LT.TOL) THEN ! EW2 is equal to EW3 D1=1.0/(EW1-EW2)/(EW1-EW3) M2=M-math_I3*EW2 M3=M-math_I3*EW3 EB1=math_mul33x33(M2,M3)*D1 EB2=math_I3-EB1 ! both EB3 and EW3 are set to zero so that they do not ! contribute to U in PDECOMPOSITION EW3=0.0 ELSE IF(C3.LT.TOL) THEN ! EW1 is equal to EW3 D2=1.0/(EW2-EW1)/(EW2-EW3) M1=M-math_I3*EW1 M3=M-math_I3*EW3 EB2=math_mul33x33(M1,M3)*D2 EB1=math_I3-EB2 ! both EB3 and EW3 are set to zero so that they do not ! contribute to U in PDECOMPOSITION EW3=0.0 ELSE ! all three eigenvectors are different D1=1.0/(EW1-EW2)/(EW1-EW3) D2=1.0/(EW2-EW1)/(EW2-EW3) D3=1.0/(EW3-EW1)/(EW3-EW2) M1=M-EW1*math_I3 M2=M-EW2*math_I3 M3=M-EW3*math_I3 EB1=math_mul33x33(M2,M3)*D1 EB2=math_mul33x33(M1,M3)*D2 EB3=math_mul33x33(M1,M2)*D3 END IF END IF RETURN END SUBROUTINE math_spectral1 !************************************************************************** ! volume of tetrahedron given by four vertices !************************************************************************** pure function math_volTetrahedron(v1,v2,v3,v4) implicit none real*8 math_volTetrahedron real*8, dimension (3), intent(in) :: v1,v2,v3,v4 real*8, dimension (3,3) :: m m(:,1) = v1-v2 m(:,2) = v2-v3 m(:,3) = v3-v4 math_volTetrahedron = math_det3x3(m)/6.0 end function math_volTetrahedron !subroutines below are for postprocessing with python !two small helper functions for indexing ! CAREFULL, index and location runs from 0 to N-1 (python style) function mesh_location(idx,resolution) integer, intent(in) :: idx integer, intent(in) :: resolution(3) integer :: mesh_location(3) mesh_location = (/modulo(idx/ resolution(3) / resolution(2),resolution(1)), & modulo(idx/ resolution(3), resolution(2)), & modulo(idx, resolution(3))/) end function mesh_location function mesh_index(location,resolution) integer, intent(in) :: location(3) integer, intent(in) :: resolution(3) integer :: mesh_index mesh_index = modulo(location(3), resolution(3)) +& (modulo(location(2), resolution(2)))*resolution(3) +& (modulo(location(1), resolution(1)))*resolution(3)*resolution(2) end function mesh_index end module math !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine mesh(res_x,res_y,res_z,geomdim,defgrad_av,centroids,nodes) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to build a regular mesh of cubes for given coordinates (= center of the cubes) ! implicit none real*8 geomdim(3) integer res_x, res_y, res_z real*8 wrappedCentroids(res_x+2,res_y+2,res_z+2,3) real*8 nodes(res_x+1,res_y+1,res_z+1,3) real*8 centroids(res_x ,res_y ,res_z ,3) integer, dimension(3,8) :: neighbor = reshape((/ & 0, 0, 0, & 1, 0, 0, & 1, 1, 0, & 0, 1, 0, & 0, 0, 1, & 1, 0, 1, & 1, 1, 1, & 0, 1, 1 & /), & (/3,8/)) integer i,j,k,n real*8, dimension(3,3) :: defgrad_av integer, dimension(3) :: diag, shift, lookup, me, res nodes = 0.0 diag = 1 shift = 0 lookup = 0 res = (/res_x,res_y,res_z/) wrappedCentroids = 0.0 wrappedCentroids(2:res_x+1,2:res_y+1,2:res_z+1,:) = centroids do k = 0,res_z+1 do j = 0,res_y+1 do i = 0,res_x+1 if (k==0 .or. k==res_z+1 .or. & ! z skin j==0 .or. j==res_y+1 .or. & ! y skin i==0 .or. i==res_x+1 ) then ! x skin me = (/i,j,k/) ! me on skin shift = sign(abs(res+diag-2*me)/(res+diag),res+diag-2*me) lookup = me-diag+shift*res wrappedCentroids(i+1,j+1,k+1,:) = centroids(lookup(1)+1,lookup(2)+1,lookup(3)+1,:) - & matmul(defgrad_av, shift*geomdim) endif enddo; enddo; enddo do k = 0,res_z do j = 0,res_y do i = 0,res_x do n = 1,8 nodes(i+1,j+1,k+1,:) = nodes(i+1,j+1,k+1,:) + wrappedCentroids(i+1+neighbor(1,n), & j+1+neighbor(2,n), & k+1+neighbor(3,n), :) enddo; enddo; enddo; enddo nodes = nodes/8.0 end subroutine mesh !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine deformed(res_x,res_y,res_z,geomdim,defgrad,defgrad_av,coord_avgCorner) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate coordinates in current configuration for given defgrad ! using linear interpolation (blurres out high frequency defomation) ! implicit none real*8 geomdim(3) integer res_x, res_y, res_z real*8 coord(8,6,res_x,res_y,res_z,3) real*8 coord_avgOrder(8,res_x,res_y,res_z,3) real*8 coord_avgCorner(res_x,res_y,res_z,3) real*8 defgrad(res_x,res_y,res_z,3,3) integer, dimension(3,8) :: corner = reshape((/ & 0, 0, 0,& 1, 0, 0,& 1, 1, 0,& 0, 1, 0,& 1, 1, 1,& 0, 1, 1,& 0, 0, 1,& 1, 0, 1 & /), & (/3,8/)) integer, dimension(3,8) :: step = reshape((/ & 1, 1, 1,& -1, 1, 1,& -1,-1, 1,& 1,-1, 1,& -1,-1,-1,& 1,-1,-1,& 1, 1,-1,& -1, 1,-1 & /), & (/3,8/)) integer, dimension(3,6) :: order = reshape((/ & 1, 2, 3,& 1, 3, 2,& 2, 1, 3,& 2, 3, 1,& 3, 1, 2,& 3, 2, 1 & /), & (/3,6/)) real*8 myStep(3), fones(3), parameter_coords(3) real*8 defgrad_av(3,3) real*8 negative(3), positive(3) integer rear(3), init(3), ones(3), oppo(3), me(3), res(3) integer i, j, k, s, o print*, 'Restore geometry using linear integration' print '(a,/,e12.5,e12.5,e12.5)', ' Dimension:', geomdim print '(a,/,i5,i5,i5)', ' Resolution:', res_x,res_y,res_z ones = 1 fones = 1.0 coord_avgOrder=0.0 res = (/res_x,res_y,res_z/) do s = 0, 7 ! corners (from 0 to 7) init = corner(:,s+1)*(res-ones) +ones oppo = corner(:,mod((s+4),8)+1)*(res-ones) +ones do o=1,6 ! orders ! from 1 to 6) do k = init(order(3,o)), oppo(order(3,o)), step(order(3,o),s+1) rear(order(2,o)) = init(order(2,o)) do j = init(order(2,o)), oppo(order(2,o)), step(order(2,o),s+1) rear(order(1,o)) = init(order(1,o)) do i = init(order(1,o)), oppo(order(1,o)), step(order(1,o),s+1) me(order(1,o)) = i me(order(2,o)) = j me(order(3,o)) = k if ( (me(1)==init(1)).and.(me(2)==init(2)).and. (me(3)==init(3)) ) then coord(s+1,o,me(1),me(2),me(3),:) = geomdim * (matmul(defgrad_av,corner(:,s+1)) + & matmul(defgrad(me(1),me(2),me(3),:,:),0.5*step(:,s+1)/res)) else myStep = (me-rear)*geomdim/res coord(s+1,o,me(1),me(2),me(3),:) = coord(s+1,o,rear(1),rear(2),rear(3),:) + & 0.5*matmul(defgrad(me(1),me(2),me(3),:,:) + & defgrad(rear(1),rear(2),rear(3),:,:),myStep) endif rear = me enddo; enddo; enddo; enddo do i=1,6 coord_avgOrder(s+1,:,:,:,:) = coord_avgOrder(s+1,:,:,:,:) + coord(s+1,i,:,:,:,:)/6.0 enddo enddo do k=0, res_z-1 do j=0, res_y-1 do i=0, res_x-1 parameter_coords = (2.0*(/i+0.0,j+0.0,k+0.0/)-real(res)+fones)/(real(res)-fones) positive = fones + parameter_coords negative = fones - parameter_coords coord_avgCorner(i+1,j+1,k+1,:) = ( coord_avgOrder(1,i+1,j+1,k+1,:) *negative(1)*negative(2)*negative(3)& + coord_avgOrder(2,i+1,j+1,k+1,:) *positive(1)*negative(2)*negative(3)& + coord_avgOrder(3,i+1,j+1,k+1,:) *positive(1)*positive(2)*negative(3)& + coord_avgOrder(4,i+1,j+1,k+1,:) *negative(1)*positive(2)*negative(3)& + coord_avgOrder(5,i+1,j+1,k+1,:) *positive(1)*positive(2)*positive(3)& + coord_avgOrder(6,i+1,j+1,k+1,:) *negative(1)*positive(2)*positive(3)& + coord_avgOrder(7,i+1,j+1,k+1,:) *negative(1)*negative(2)*positive(3)& + coord_avgOrder(8,i+1,j+1,k+1,:) *positive(1)*negative(2)*positive(3))*0.125 enddo; enddo; enddo end subroutine deformed !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine deformed_fft(res_x,res_y,res_z,geomdim,defgrad,defgrad_av,scaling,coords) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate coordinates in current configuration for given defgrad ! using integration in Fourier space (more accurate than deformed(...)) ! implicit none integer res_x, res_y, res_z real*8 geomdim(3) real*8 defgrad(res_x,res_y,res_z,3,3) real*8 defgrad_av(3,3) real*8 scaling real*8 coords(res_x,res_y,res_z,3) complex*16 coords_fft(res_x/2+1,res_y,res_z,3) complex*16 defgrad_fft(res_x,res_y,res_z,3,3) integer i, j, k integer k_s(3) real*8 step(3) real*8 offset_coords(3) real*8, parameter :: pi = 3.14159265358979323846264338327950288419716939937510 integer*8 :: plan_fft(2) include 'fftw3.f' !header file for fftw3 (declaring variables). Library files are also needed print*, 'Restore geometry using FFT-based integration' print '(a,/,e12.5,e12.5,e12.5)', ' Dimension:', geomdim print '(a,/,i5,i5,i5)', ' Resolution:', res_x,res_y,res_z call dfftw_plan_many_dft(plan_fft(1),3,(/res_x,res_y,res_z/),9,& defgrad_fft,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,& defgrad_fft,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,FFTW_FORWARD,FFTW_PATIENT) call dfftw_plan_many_dft_c2r(plan_fft(2),3,(/res_x,res_y,res_z/),3,& coords_fft,(/res_x/2+1,res_y,res_z/),1,(res_x/2+1)*res_y*res_z,& coords, (/res_x, res_y,res_z/),1, res_x* res_y*res_z,FFTW_PATIENT) coords_fft = 0.0 defgrad_fft = defgrad step(1) = geomdim(1)/real(res_x) step(2) = geomdim(2)/real(res_y) step(3) = geomdim(3)/real(res_z) call dfftw_execute_dft(plan_fft(1), defgrad_fft, defgrad_fft) do k = 1, res_z k_s(3) = k-1 if(k > res_z/2+1) k_s(3) = k_s(3)-res_z do j = 1, res_y k_s(2) = j-1 if(j > res_y/2+1) k_s(2) = k_s(2)-res_y do i = 1, res_x/2+1 k_s(1) = i-1 if(i/=1) coords_fft(i,j,k,:) = coords_fft(i,j,k,:)& + defgrad_fft(i,j,k,:,1)*geomdim(1)/(real(k_s(1))*cmplx(0.0,1.0)*pi*2.0) if(j/=1) coords_fft(i,j,k,:) = coords_fft(i,j,k,:)& + defgrad_fft(i,j,k,:,2)*geomdim(2)/(real(k_s(2))*cmplx(0.0,1.0)*pi*2.0) if(k/=1) coords_fft(i,j,k,:) = coords_fft(i,j,k,:)& + defgrad_fft(i,j,k,:,3)*geomdim(3)/(real(k_s(3))*cmplx(0.0,1.0)*pi*2.0) enddo; enddo; enddo call dfftw_execute_dft_c2r(plan_fft(2), coords_fft, coords) coords = coords/real(res_x*res_y*res_z) offset_coords = matmul(defgrad(1,1,1,:,:),step/2.0) - scaling*coords(1,1,1,:) do k = 1, res_z; do j = 1, res_y; do i = 1, res_x coords(i,j,k,:) = scaling*coords(i,j,k,:) + offset_coords + matmul(defgrad_av,& (/step(1)*real(i-1),& step(2)*real(j-1),& step(3)*real(k-1)/)) enddo; enddo; enddo end subroutine deformed_fft !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine volume_compare(res_x,res_y,res_z,geomdim,nodes,defgrad,volume_mismatch) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate the mismatch between volume of reconstructed (compatible) ! cube and determinant of defgrad at the FP use math implicit none real*8 geomdim(3) integer res_x, res_y, res_z real*8 nodes(res_x+1,res_y+1,res_z+1,3) real*8 defgrad(res_x ,res_y ,res_z ,3,3) real*8 volume_mismatch(res_x ,res_y ,res_z ) real*8 coords(8,3) integer i,j,k real*8 vol_initial print*, 'Calculating volume mismatch' vol_initial = geomdim(1)*geomdim(2)*geomdim(3)/real(res_x)/real(res_y)/real(res_z) do k = 1,res_z do j = 1,res_y do i = 1,res_x coords(1,:) = nodes(i ,j ,k ,:) coords(2,:) = nodes(i+1,j ,k ,:) coords(3,:) = nodes(i+1,j+1,k ,:) coords(4,:) = nodes(i ,j+1,k ,:) coords(5,:) = nodes(i ,j, k+1,:) coords(6,:) = nodes(i+1,j ,k+1,:) coords(7,:) = nodes(i+1,j+1,k+1,:) coords(8,:) = nodes(i ,j+1,k+1,:) volume_mismatch(i,j,k) = abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(8,:),coords(4,:))) & + abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(8,:),coords(5,:))) & + abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(3,:),coords(4,:))) & + abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(3,:),coords(2,:))) & + abs(math_volTetrahedron(coords(7,:),coords(5,:),coords(2,:),coords(6,:))) & + abs(math_volTetrahedron(coords(7,:),coords(5,:),coords(2,:),coords(1,:))) volume_mismatch(i,j,k) = volume_mismatch(i,j,k)/math_det3x3(defgrad(i,j,k,:,:)) enddo; enddo; enddo volume_mismatch = volume_mismatch/vol_initial end subroutine volume_compare !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine shape_compare(res_x,res_y,res_z,geomdim,nodes,centroids,defgrad,shape_mismatch) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate the mismatch between the vectors from the central point to ! the corners of reconstructed (combatible) volume element and the vectors calculated by deforming ! the initial volume element with the current deformation gradient implicit none real*8 geomdim(3) integer res_x, res_y, res_z real*8 nodes(res_x+1,res_y+1,res_z+1,3) real*8 centroids(res_x ,res_y ,res_z ,3) real*8 defgrad(res_x ,res_y ,res_z ,3,3) real*8 shape_mismatch(res_x ,res_y ,res_z) real*8 coords_initial(8,3) integer i,j,k print*, 'Calculating shape mismatch' coords_initial(1,:) = (/-geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/) coords_initial(2,:) = (/+geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/) coords_initial(3,:) = (/+geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/) coords_initial(4,:) = (/-geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/) coords_initial(5,:) = (/-geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/) coords_initial(6,:) = (/+geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/) coords_initial(7,:) = (/+geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/) coords_initial(8,:) = (/-geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/) do i=1,8 enddo do k = 1,res_z do j = 1,res_y do i = 1,res_x shape_mismatch(i,j,k) = & sqrt(sum((nodes(i ,j ,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(1,:)))**2.0))& + sqrt(sum((nodes(i+1,j ,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(2,:)))**2.0))& + sqrt(sum((nodes(i+1,j+1,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(3,:)))**2.0))& + sqrt(sum((nodes(i ,j+1,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(4,:)))**2.0))& + sqrt(sum((nodes(i ,j, k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(5,:)))**2.0))& + sqrt(sum((nodes(i+1,j ,k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(6,:)))**2.0))& + sqrt(sum((nodes(i+1,j+1,k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(7,:)))**2.0))& + sqrt(sum((nodes(i ,j+1,k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(8,:)))**2.0)) enddo; enddo; enddo end subroutine shape_compare !++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine inverse_reconstruction(res_x,res_y,res_z,reference_configuration,current_configuration,defgrad) !++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate deformation gradient from reference and current configuration ! NOT WORKING BY NOW!!!!!!!!!!!!! ! use math implicit none integer res_x, res_y, res_z real*8 reference_configuration(res_x+1,res_y+1,res_z+1,3) real*8 current_configuration(res_x+1,res_y+1,res_z+1,3) real*8 defgrad(res_x,res_y,res_z,3,3) real*8 delta, tolerance, res, res_center real*8 reference(8,3) real*8 current(8,3) real*8 defgrad_temp(3,3) real*8 dres_dF(3,3) real*8 identity(3,3) real*8 ref_bar(3) real*8 current_bar(3) real*8 r(8) real*8 differentiate(9,3,3) integer i, j, k, m, l, x, y, o identity = 0.0 identity(1,1) = 1.0 identity(2,2) = 1.0 identity(3,3) = 1.0 differentiate = 0.0 tolerance = 1e-10 delta = 1e-9 k = 0 do j = 1, 3; do i = 1, 3 k = k+1 differentiate(k,i,j) = 1.0 enddo; enddo do k = 1, res_z do j = 1, res_y do i = 1, res_x reference(1,:) = reference_configuration(i ,j ,k ,:) reference(2,:) = reference_configuration(i+1,j ,k ,:) reference(3,:) = reference_configuration(i+1,j+1,k ,:) reference(4,:) = reference_configuration(i ,j+1,k ,:) reference(5,:) = reference_configuration(i ,j ,k+1,:) reference(6,:) = reference_configuration(i+1,j ,k+1,:) reference(7,:) = reference_configuration(i+1,j+1,k+1,:) reference(8,:) = reference_configuration(i ,j+1,k+1,:) current(1,:) = current_configuration(i ,j ,k ,:) current(2,:) = current_configuration(i+1,j ,k ,:) current(3,:) = current_configuration(i+1,j+1,k ,:) current(4,:) = current_configuration(i ,j+1,k ,:) current(5,:) = current_configuration(i ,j ,k+1,:) current(6,:) = current_configuration(i+1,j ,k+1,:) current(7,:) = current_configuration(i+1,j+1,k+1,:) current(8,:) = current_configuration(i ,j+1,k+1,:) do o=1,3 ref_bar(o) = sum(reference(:,o))/8.0 current_bar(o) = sum(current(:,o))/8.0 enddo do o=1,8 reference(o,:) = reference(o,:) -ref_bar current(o,:) = current(o,:) -current_bar enddo defgrad_temp = identity res_center = 2.0*tolerance o=0 do while(res_center >= tolerance) o = o + 1 do l = 1,8 ! loop over corners r(l) = sqrt(sum((current(l,:)-matmul(defgrad_temp,reference(l,:)))**2)) ! corner distance enddo res_center = sum(r*r) ! current residuum print*, 'res_center', res_center m=0 do y=1,3; do x=1,3 ! numerical differentiation m = m+1 do l = 1,8 r(l) = sqrt(sum((current(l,:)-matmul((defgrad_temp+differentiate(m,:,:)*delta),reference(l,:)))**2)) ! corner distance enddo res = sum(r*r) print*,'res step', m, res dres_dF(x,y) = (res-res_center)/delta enddo; enddo print*, 'dres_dF', dres_dF print*, 'deltadef', math_inv3x3(dres_dF)*res_center defgrad_temp = defgrad_temp - math_inv3x3(dres_dF)*res_center ! Newton--Raphson print*, o, res_center ! pause enddo defgrad(i,j,k,:,:) = defgrad_temp enddo; enddo; enddo end subroutine inverse_reconstruction !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine tensor_avg(res_x,res_y,res_z,tensor,avg) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate average of tensor field ! implicit none integer res_x, res_y, res_z real*8 tensor(res_x,res_y,res_z,3,3) real*8 avg(3,3) real*8 wgt integer m,n wgt = 1/real(res_x*res_y*res_z) do m = 1,3; do n = 1,3 avg(m,n) = sum(tensor(:,:,:,m,n)) * wgt enddo; enddo end subroutine tensor_avg !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine logstrain_spat(res_x,res_y,res_z,defgrad,logstrain_field) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate logarithmic strain in spatial configuration for given defgrad field ! use math implicit none integer res_x, res_y, res_z integer i, j, k real*8 defgrad(res_x,res_y,res_z,3,3) real*8 logstrain_field(res_x,res_y,res_z,3,3) real*8 temp33_Real(3,3), temp33_Real2(3,3) real*8 eigenvectorbasis(3,3,3) real*8 eigenvalue(3) logical errmatinv do k = 1, res_z; do j = 1, res_y; do i = 1, res_x call math_pDecomposition(defgrad(i,j,k,:,:),temp33_Real2,temp33_Real,errmatinv) !store R in temp33_Real temp33_Real2 = math_inv3x3(temp33_Real) temp33_Real = math_mul33x33(defgrad(i,j,k,:,:),temp33_Real2) ! v = F o inv(R), store in temp33_Real2 call math_spectral1(temp33_Real, eigenvalue(1), eigenvalue(2), eigenvalue(3),& eigenvectorbasis(1,:,:), eigenvectorbasis(2,:,:), eigenvectorbasis(3,:,:)) eigenvalue = log(sqrt(eigenvalue)) logstrain_field(i,j,k,:,:) = eigenvalue(1)*eigenvectorbasis(1,:,:)+& eigenvalue(2)*eigenvectorbasis(2,:,:)+& eigenvalue(3)*eigenvectorbasis(3,:,:) enddo; enddo; enddo end subroutine logstrain_spat !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine logstrain_mat(res_x,res_y,res_z,defgrad,logstrain_field) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate logarithmic strain in material configuration for given defgrad field ! use math implicit none integer res_x, res_y, res_z integer i, j, k real*8 defgrad(res_x,res_y,res_z,3,3) real*8 logstrain_field(res_x,res_y,res_z,3,3) real*8 temp33_Real(3,3), temp33_Real2(3,3) real*8 eigenvectorbasis(3,3,3) real*8 eigenvalue(3) logical errmatinv do k = 1, res_z; do j = 1, res_y; do i = 1, res_x call math_pDecomposition(defgrad(i,j,k,:,:),temp33_Real,temp33_Real2,errmatinv) !store U in temp33_Real call math_spectral1(temp33_Real, eigenvalue(1), eigenvalue(2), eigenvalue(3),& eigenvectorbasis(1,:,:), eigenvectorbasis(2,:,:), eigenvectorbasis(3,:,:)) eigenvalue = log(sqrt(eigenvalue)) logstrain_field(i,j,k,:,:) = eigenvalue(1)*eigenvectorbasis(1,:,:)+& eigenvalue(2)*eigenvectorbasis(2,:,:)+& eigenvalue(3)*eigenvectorbasis(3,:,:) enddo; enddo; enddo end subroutine logstrain_mat !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine calculate_cauchy(res_x,res_y,res_z,defgrad,p_stress,c_stress) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate cauchy stress for given PK1 stress and defgrad field ! use math implicit none integer res_x, res_y, res_z integer i, j, k real*8 defgrad(res_x,res_y,res_z,3,3) real*8 p_stress(res_x,res_y,res_z,3,3) real*8 c_stress(res_x,res_y,res_z,3,3) real*8 jacobi c_stress = 0.0 do k = 1, res_z; do j = 1, res_y; do i = 1, res_x jacobi = math_det3x3(defgrad(i,j,k,:,:)) c_stress(i,j,k,:,:) = matmul(p_stress(i,j,k,:,:),transpose(defgrad(i,j,k,:,:)))/jacobi enddo; enddo; enddo end subroutine calculate_cauchy !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine calculate_mises(res_x,res_y,res_z,tensor,vm) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate von Mises equivalent of tensor field ! implicit none integer res_x, res_y, res_z integer i, j, k real*8 tensor(res_x,res_y,res_z,3,3) real*8 vm(res_x,res_y,res_z,1) real*8 deviator(3,3) real*8 delta(3,3) real*8 J_2 delta =0.0 delta(1,1) = 1.0 delta(2,2) = 1.0 delta(3,3) = 1.0 do k = 1, res_z; do j = 1, res_y; do i = 1, res_x deviator = tensor(i,j,k,:,:) - 1.0/3.0*tensor(i,j,k,1,1)*tensor(i,j,k,2,2)*tensor(i,j,k,3,3)*delta J_2 = deviator(1,1)*deviator(2,2)& + deviator(2,2)*deviator(3,3)& + deviator(1,1)*deviator(3,3)& - (deviator(1,2))**2& - (deviator(2,3))**2& - (deviator(1,3))**2 vm(i,j,k,:) = sqrt(3*J_2) enddo; enddo; enddo end subroutine calculate_mises !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine divergence_fft(res_x,res_y,res_z,vec_tens,geomdim,field,divergence_field) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! calculates divergence field using integration in Fourier space !use vec_tens to decide if tensor (3) or vector (1) implicit none integer res_x, res_y, res_z, vec_tens real*8 geomdim(3) real*8 field(res_x,res_y,res_z,vec_tens,3) real*8 field_copy(res_x,res_y,res_z,vec_tens,3) real*8 xi(res_x,res_y,res_z,3) real*8 divergence_field(res_x,res_y,res_z,vec_tens) complex*16 divergence_field_fft(res_x/2+1,res_y,res_z,vec_tens) complex*16 field_fft(res_x,res_y,res_z,vec_tens,3) complex*16 img integer i, j, k real*8, parameter :: pi = 3.14159265358979323846264338327950288419716939937510 integer*8 :: plan_fft(2) include 'fftw3.f' !header file for fftw3 (declaring variables). Library files are also needed img = cmplx(0.0,1.0) call dfftw_plan_many_dft_r2c(plan_fft(1),3,(/res_x,res_y,res_z/),vec_tens*3,& field_copy,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,& field_fft,(/res_x/2+1,res_y,res_z/),1,(res_x/2+1)*res_y*res_z,FFTW_PATIENT) call dfftw_plan_many_dft_c2r(plan_fft(2),3,(/res_x,res_y,res_z/),vec_tens,& divergence_field_fft,(/res_x/2+1,res_y,res_z/),1,(res_x/2+1)*res_y*res_z,& divergence_field,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,FFTW_PATIENT) ! field_copy is destroyed during plan creation field_copy = field call dfftw_execute_dft_r2c(plan_fft(1), field_copy, field_fft) xi = 0.0 ! Alternative calculation of discrete frequencies k_s, ordered as in FFTW (wrap around) ! do k = 0,res_z/2 -1 ! do j = 0,res_y/2 -1 ! do i = 0,res_x/2 -1 ! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/-i,-j,-k/)/geomdim ! xi(1+i, 1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/ i,-j,-k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+j, 1+mod(res_z-k,res_z),:) = (/-i, j,-k/)/geomdim ! xi(1+i, 1+j, 1+mod(res_z-k,res_z),:) = (/ i, j,-k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+k, :) = (/-i,-j, k/)/geomdim ! xi(1+i, 1+mod(res_y-j,res_y),1+k, :) = (/ i,-j, k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+j, 1+k, :) = (/-i, j, k/)/geomdim ! xi(1+i, 1+j, 1+k, :) = (/ i, j, k/)/geomdim ! xi(1+i, 1+j, 1+k, :) = (/ i, j, k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+j, 1+k, :) = (/-i, j, k/)/geomdim ! xi(1+i, 1+mod(res_y-j,res_y),1+k, :) = (/ i,-j, k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+k, :) = (/-i,-j, k/)/geomdim ! xi(1+i, 1+j, 1+mod(res_z-k,res_z),:) = (/ i, j,-k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+j, 1+mod(res_z-k,res_z),:) = (/-i, j,-k/)/geomdim ! xi(1+i, 1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/ i,-j,-k/)/geomdim ! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/-i,-j,-k/)/geomdim ! enddo; enddo; enddo do k = 0, res_z-1 do j = 0, res_y-1 do i = 0, res_x/2 xi(i+1,j+1,k+1,:) = (/real(i),real(j),real(k)/)/geomdim if(k==res_z/2) xi(i+1,j+1,k+1,3)= 0.0 ! set highest frequencies to zero if(j==res_y/2) xi(i+1,j+1,k+1,2)= 0.0 if(i==res_x/2) xi(i+1,j+1,k+1,1)= 0.0 enddo; enddo; enddo do k = 1, res_z do j = 1, res_y do i = 1, res_x/2+1 divergence_field_fft(i,j,k,1) = sum(field_fft(i,j,k,1,:)*xi(i,j,k,:)) if(vec_tens == 3) then divergence_field_fft(i,j,k,2) = sum(field_fft(i,j,k,2,:)*xi(i,j,k,:)) divergence_field_fft(i,j,k,3) = sum(field_fft(i,j,k,3,:)*xi(i,j,k,:)) endif enddo; enddo; enddo divergence_field_fft = divergence_field_fft*img*2.0*pi call dfftw_execute_dft_c2r(plan_fft(2), divergence_field_fft, divergence_field) end subroutine divergence_fft !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine divergence(res_x,res_y,res_z,vec_tens,order,geomdim,field,divergence_field) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! calculates divergence field using FDM with variable accuracy !use vec_tes to decide if tensor (3) or vector (1) use math implicit none integer res_x, res_y, res_z, vec_tens, order integer coordinates(6,3) real*8 geomdim(3) real*8 field(res_x,res_y,res_z,vec_tens,3) real*8 divergence_field(res_x,res_y,res_z,vec_tens) integer i, j, k, m, l real*8, dimension(4,4) :: FDcoefficient = reshape((/ & !from http://en.wikipedia.org/wiki/Finite_difference_coefficients 1.0/2.0, 0.0, 0.0, 0.0,& 2.0/3.0,-1.0/12.0, 0.0, 0.0,& 3.0/4.0,-3.0/20.0,1.0/ 60.0, 0.0,& 4.0/5.0,-1.0/ 5.0,4.0/105.0,-1.0/280.0/),& (/4,4/)) divergence_field = 0.0 order = order + 1 do k = 0, res_z-1; do j = 0, res_y-1; do i = 0, res_x-1 do m = 1, order coordinates(1,:) = mesh_location(mesh_index((/i+m,j,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/) coordinates(2,:) = mesh_location(mesh_index((/i-m,j,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/) coordinates(3,:) = mesh_location(mesh_index((/i,j+m,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/) coordinates(4,:) = mesh_location(mesh_index((/i,j-m,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/) coordinates(5,:) = mesh_location(mesh_index((/i,j,k+m/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/) coordinates(6,:) = mesh_location(mesh_index((/i,j,k-m/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/) do l = 1, vec_tens divergence_field(i+1,j+1,k+1,l) = divergence_field(i+1,j+1,k+1,l) + FDcoefficient(m,order) * & ((field(coordinates(1,1),coordinates(1,2),coordinates(1,3),l,1)- & field(coordinates(2,1),coordinates(2,2),coordinates(2,3),l,1))*real(res_x)/geomdim(1) +& (field(coordinates(3,1),coordinates(3,2),coordinates(3,3),l,2)- & field(coordinates(4,1),coordinates(4,2),coordinates(4,3),l,2))*real(res_y)/geomdim(2) +& (field(coordinates(5,1),coordinates(5,2),coordinates(5,3),l,3)- & field(coordinates(6,1),coordinates(6,2),coordinates(6,3),l,3))*real(res_z)/geomdim(3)) enddo enddo enddo; enddo; enddo end subroutine divergence